Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 4.
WEIGHTED SHIFTS
above mentioned notions behave under finite perturbations of the weight sequence.
We first obtain three concrete results:
(i) the subnormality of W α is never stable under nonzero finite rank perturbations
unless the perturbation is confined to the zeroth weight;
(ii) 2-hyponormality implies positive quadratic hyponormality, in the sense that
the Maclaurin coefficients of D n (s) := det P n [(W α + sWα) 2 ∗ , W α + sWα] 2 P n are nonnegative,
for every n ≥ 0, where P n denotes the orthogonal projection onto the basis
vectors {e 0 , · · · , e n }; and
(iii) if α is strictly increasing and W α is 2-hyponormal then for α ′ a small perturbation
of α, the shift W α ′ remains positively quadratically hyponormal.
Along the way we establish two related results, each of independent interest:
(iv) an integrality criterion for a subnormal weighted shift to have an n-step subnormal
extension; and
(v) a proof that the sets of k-hyponormal and weakly k-hyponormal operators are
closed in the strong operator topology.
C. Berger’s characterization of subnormality for unilateral weighted shifts states
that W α is subnormal if and only if there exists a Borel probability measure µ supported
in [0, ||W α || 2 ], with ||W α || 2 ∈ supp µ, such that
∫
γ n = t n dµ(t) for all n ≥ 0.
Given an initial segment of weights α : α 0 , · · · α m , the sequence ̂α ∈ l ∞ (Z + ) such
that ̂α : α i (i = 0, · · · , m) is said to be recursively generated by α if there exists r ≥ 1
and φ 0 , · · · , φ r−1 ∈ R such that
γ n+r = φ 0 γ n + · · · + φ r−1 γ n+r−1 (all n ≥ 0),
where γ 0 = 1, γ n = α 2 0 · · · α 2 n−1 (n ≥ 1). In this case, Ŵα with weights ̂α is said to
be recursively generated. If we let
g(t) := t r − (φ r−1 t r−1 + · · · + φ 0 )
then g has r distinct real roots 0 ≤ s 0 < · · · < s r−1 . Then Ŵα is a subnormal shift
whose Berger measure µ is given by
µ = ρ 0 δ s0 + · · · + ρ r−1 δ sr−1,
where (ρ 0 , · · · , ρ r−1 ) is the unique solution of the Vandermonde equation
⎡
⎤ ⎡ ⎤ ⎡ ⎤
1 1 · · · 1 ρ 0 γ 0
s 0 s 1 · · · s r−1
ρ 1
⎢
⎥ ⎢
⎣ . . . ⎦ ⎣
⎥
. ⎦ = γ 1
⎢
⎣
⎥
. ⎦ .
s r−1
0 s r−1
1 · · · s r−1
r−1
ρ r−1 γ r−1
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